This paper revisits the problem of multi-server Private Information Retrieval with Private Side Information (PIR-PSI). In this problem, $N$ non-colluding servers store identical copies of $K$ messages, each comprising $L$ symbols from $\mathbb{F}_q$, and a user, who knows $M$ of these messages, wants to retrieve one of the remaining $K-M$ messages. The user's goal is to retrieve the desired message by downloading the minimum amount of information from the servers while revealing no information about the identities of the desired message and side information messages to any server. The capacity of PIR-PSI, defined as the maximum achievable download rate, was previously characterized for all $N$, $K$, and $M$ when $L$ and $q$ are sufficiently large -- specifically, growing exponentially with $K$, to ensure the divisibility of each message into $N^K$ sub-packets and to guarantee the existence of an MDS code with its length and dimension being exponential in $K$. In this work, we propose a new capacity-achieving PIR-PSI scheme that is applicable to all $N$, $K$, $M$, $L$, and $q$ where $N\geq M+1$ and $N-1\mid L$. The proposed scheme operates with a sub-packetization level of $N-1$, independent of $K$, and works over any finite field without requiring an MDS code.

翻译：本文重新审视了多服务器环境下具有私密边信息的私密信息检索（PIR-PSI）问题。在该问题中，$N$个非共谋服务器存储了$K$条消息的相同副本，每条消息包含$\mathbb{F}_q$上的$L$个符号；用户已知其中的$M$条消息，并希望检索剩余$K-M$条消息中的一条。用户的目标是通过从服务器下载最少的信息量来获取目标消息，同时不向任何服务器泄露目标消息和边信息消息的身份信息。PIR-PSI的容量定义为最大可达下载速率，此前已在$L$和$q$足够大（即随$K$呈指数增长，以确保每条消息可划分为$N^K$个子包，并保证存在一个长度和维度均随$K$指数增长的最大距离可分码）的条件下，针对所有$N$、$K$和$M$进行了刻画。本文提出了一种新的容量可达PIR-PSI方案，该方案适用于所有满足$N\geq M+1$且$N-1\mid L$的$N$、$K$、$M$、$L$和$q$。所提方案的子包化水平为$N-1$，与$K$无关，且可在无需MDS码的情况下在任何有限域上工作。